Binomial Gaussian

Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution cur ves. It is most unlikely that any frequency distribution cur ve obtained in practice will exactly fit a cur ve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

As a graphic summary of the properties of the three distributions, illustrative binomial, Poisson, and Gaussian distributions (the latter unique for the N chosen) have been plotted in Figure 10-2. 

The median, like the mean, is a measure of the midpoint or center of the distribution. In some cases, such as the gaussian or uniform distribution, the median coincides with the mean, but consideration of the binomial distribution with p i shows that this is not generally true. The... 

Determine the copolymer composition for a styrene-acrylonitrile copolymer made at the azeotrope (62 mol% styrene). Assume = 1000. One approach is to use the Gaussian approximation to the binomial distribution. Another is to synthesize 100,000 or so molecules using a random number generator and to sort them by composition. 

The random-walk model of diffusion can also be applied to derive the shape of the bell-shaped concentration profile characteristic of bulk diffusion. As in the previous section, a planar layer of N tracer atoms is the starting point. Each atom diffuses from the interface by a random walk of n steps in a direction perpendicular to the interface. As mentioned (see footnote 5) the statistics are well known and described by the binomial distribution (Fig. S5.5a-S5.5c). At large values of N, this discrete distribution can be approximated by a continuous function, the Gaussian distribution curve7 with a form ... 

If the distribution is Gaussian or binomial, the minimum number of increments can be e tijated from... 

A further simplification of the parent binomial distribution occurs when the number of successes is relatively large, that is, we get more than about 30 counts in a measurement. Then, the binomial distribution can be represented as a normal or Gaussian distribution. Here we write... 

Fig. 9.1. (A) Gaussian (a) and sine (b) excitation profiles. (B) Composite (G3) Gaussian pulse. (C) Train of soft pulses modified after the DANTE sequence to achieve selective off-resonance excitation. (D) Redfield 21412 sequence. (E) Binomial 11, 121, 1331, 14641 sequences. (F) JR (a) and compensated JR (or 1111) (b) sequences. (G) Watergate sequence. (H) Weft (Superweft) sequence. (I) Modeft sequence. (J) MLEV16 sequence. (K) NOESY sequence with trim pulse. (L) MLEV17 sequence with trim pulses. (M) Clean-TOCSY sequence.

Examples of distributions are the binomial and the POISSON distribution and the GAUSSian (normal), y2-, t-, and -distributions. 

Equation 4.7 is very difficult to use for even a few hundred random events because factorials grow rapidly. However, the agreement between the binomial values and a Gaussian curve in Figure 4.1 is not accidental. It can be shown that if TV +> 1 and M <+ TV,... 

Just as with the binomial distribution, calculating factorials is tedious for large N. The binomial distribution converged to a Gaussian for large N (Equation 4.11). The most probable distribution for the multinomial expansion converges to an exponential ... 

Fig. 14.9 (a) The efflux rate, (b) Binomial (dashed lines) and gaussian (solid lines) distributions for the cleavage probability. 

It can be shown that for large n, the binomial distribution becomes the Gaussian distribution (see Problem 2.21.1). 

The variation that is observed in experimental results can take many different forms or distributions. We consider here three of the best known that can be expressed in relatively straightforward mathematical terms the binomial distribution, the Poisson distribution and the Gaussian, or normal, distribution. These are all forms of parametric statistics which are based on the idea that the data are spread in a specific manner. Ideally, this should be demonstrated before a statistical analysis is carried out, but this is not often done. 

In the Poisson and binomial distributions, the mean and variance are not independent quantities, and in the Poisson distribution they are equal. This is not an appropriate description of most measurements or observations, where the variance depends on the type of experiment. For example, a series of repeated weighings of an object will give an average value, but the spread of the observed values will depend on the quality and precision of the balance used. In other words, the mean and variance are independent quantities, and different two parameter statistical distribution functions are needed to describe these situations. The most celebrated such function is the Gaussian, or normal, distribution ... 

Martin and Synge pointed out that the successive terms of the binomial expansion approximate more and more closely the gaussian distribution of statistics (Section 26-3) as the number of terms increases. Craig wrote for the gaussian curve... 

Equation (23-24) is an example of a Poisson distribution, characteristic of a continuous-flow process, in contrast to the binomial-distribution characteristic of a batch process [Equation (23-13)]. Both approximate the gaussian-distribution curve when the number of stages becomes large. ... 

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